Rho

Buying a call is a way of controlling an asset without paying for it yet. The money you did not spend sits earning interest until expiry. The higher that interest rate, the more that deferral is worth, and the more the call is worth.

A put is the mirror. Holding a put means you will receive the strike price at expiry rather than selling today; the higher the interest rate, the less that deferred receipt is worth in today's money, and the less the put is worth.

Rho puts a number on both. The workbook states the direction plainly: high interest rates increase the value of a call option and decrease the value of a put option. It is the least influential of the five Greeks in ordinary conditions, and the one that matters most on long-dated contracts and in a rate cycle.

The mechanism is visible in the put-call-parity relation, which the workbook gives in full:

c + X·e^(−rt) = p + S

The only place the interest rate appears is in the term X·e^(−rt), the present value of the strike. Raise r and that present value falls. Since the spot S has not moved, the left side must fall too — so c − p must rise. The call gains exactly what the put loses, and the size of the shift is the change in the discounted strike.

Three consequences follow:

• Rho grows with time to expiry. Discounting one month of interest barely moves a price; discounting a year moves it visibly. A weekly option has almost no rho.
• Rho grows with the strike, because the strike is the thing being discounted. Deep in-the-money options carry the largest rho.
• Rho is the smallest of the Greeks in normal markets. It matters when rates move a lot, or when the contract is long-dated — which is why it comes into its own in interest rate derivatives rather than in weekly index options.

Rho = Change in option premium ÷ Change in cost of funding the underlying

And, from put-call parity, the aggregate effect of a rate change on the call-minus-put spread:

c − p = S − X·e^(−rt)

Δ(c − p) = X·e^(−r₁t) − X·e^(−r₂t)        for a rate move from r₁ to r₂

A one-month European option pair is struck at X = 17,500, with a contract size of 50. The risk-free rate is 6% p.a. and t = 1/12.

Discounted strike at 6%:

17,500 × e^(−0.06 × 1/12) = 17,500 × 0.995012 = Rs 17,412.72

The RBI raises rates and the one-month rate moves to 7%:

17,500 × e^(−0.07 × 1/12) = 17,500 × 0.994184 = Rs 17,398.21

The present value of the strike has fallen by Rs 14.51. Put-call parity forces the whole of that into the call-minus-put spread:

Δ(c − p) = +14.51 index points
Per contract = 14.51 × 50 = Rs 725.50

So across a hundred lots of a call-versus-put book, a one-percentage-point move in the one-month rate is worth about Rs 72,550, entirely independent of where the index goes.

Now stretch the maturity to one year, same strike, same rate move:

17,500 × e^(−0.06) = Rs 16,480.87
17,500 × e^(−0.07) = Rs 16,316.90
Δ(c − p)           = Rs 163.97 → Rs 8,198.50 per contract

Eleven times the effect, for the same one-point rate move. That is the entire lesson of rho: on a weekly index option it is a rounding error, and on a long-dated contract it is the second-largest number on the sheet.

Chapter 16.7 lists rho among the five Greeks and defines it as the change in an option's price per unit increase in the cost of funding the underlying; the "Interest Rates" sub-section of the same chapter supplies the direction — calls up, puts down. Expect a matching question across the Greeks and a direction question on what a rate rise does to a put premium. The arithmetic above is derived from the workbook's own put-call parity formula in Chapter 17.3.

• Rho is positive for calls and negative for puts. This is the one Greek whose sign genuinely differs by contract type for the buyer.
• The workbook notes that interest rates affect stock and index options more than options on futures. Do not generalise the direction across every underlying.
• Rho is small on short-dated contracts. On a weekly index option it is almost invisible; the exam still expects the direction.
• Do not confuse the funding rate with the dividend yield. They pull opposite ways, and the workbook's Black-Scholes treatment explicitly ignores dividends.
• Rho measures a one percentage point move, not a 1% relative move. From 6% to 7%, not from 6% to 6.06%.
• Rho is the reason a rate cut is not automatically good for options. It lifts puts and drags on calls — the opposite of the reflex answer.